On ergodic properties of some Levy-type processes

In this note we prove some sufficient conditions for ergodicity of a Levy-type process, such that on the test functions the generator of the respective semigroup is of the form $$Lf(x) = a(x)f'(x) + \int_{\mathbb{R}}{ \left( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb{I}_{|u|\leq 1} \right) \nu(x,du)}, \quad f\in C_{\infty}^{2}(\mathbb{R}).$$ Here $\nu(x,du)$ is a Levy-type kernel and $a(\cdot): \mathbb{R}\to \mathbb{R}$. We consider the case when the tails are of polynomial decay as well as the case when the decay is (sub)-exponential. For the proof the Foster-Lyapunov approach is used.